↑

Properties

Label 2-490-245.239-c1-0-14
Degree $2$
Conductor $490$
Sign $0.770 - 0.637i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.781 + 0.623i)2-s + (0.0985 − 0.204i)3-s + (0.222 + 0.974i)4-s + (−0.593 − 2.15i)5-s + (0.204 − 0.0985i)6-s + (0.719 + 2.54i)7-s + (−0.433 + 0.900i)8-s + (1.83 + 2.30i)9-s + (0.880 − 2.05i)10-s + (1.14 − 1.43i)11-s + (0.221 + 0.0505i)12-s + (0.208 + 0.166i)13-s + (−1.02 + 2.43i)14-s + (−0.499 − 0.0909i)15-s + (−0.900 + 0.433i)16-s + (3.36 + 0.768i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.0568 − 0.118i)3-s + (0.111 + 0.487i)4-s + (−0.265 − 0.964i)5-s + (0.0835 − 0.0402i)6-s + (0.271 + 0.962i)7-s + (−0.153 + 0.318i)8-s + (0.612 + 0.768i)9-s + (0.278 − 0.650i)10-s + (0.345 − 0.433i)11-s + (0.0639 + 0.0145i)12-s + (0.0577 + 0.0460i)13-s + (−0.273 + 0.651i)14-s + (−0.128 − 0.0234i)15-s + (−0.225 + 0.108i)16-s + (0.817 + 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92578 + 0.693197i\)
\(L(\frac12)\) \(\approx\) \(1.92578 + 0.693197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
5 \( 1 + (0.593 + 2.15i)T \)
7 \( 1 + (-0.719 - 2.54i)T \)
good3 \( 1 + (-0.0985 + 0.204i)T + (-1.87 - 2.34i)T^{2} \)
11 \( 1 + (-1.14 + 1.43i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.208 - 0.166i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-3.36 - 0.768i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 8.08T + 19T^{2} \)
23 \( 1 + (5.46 - 1.24i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.12 + 4.94i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 - 5.34T + 31T^{2} \)
37 \( 1 + (4.72 + 1.07i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (2.20 + 1.06i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-2.42 - 5.03i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (6.10 + 4.86i)T + (10.4 + 45.8i)T^{2} \)
53 \( 1 + (9.46 - 2.16i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (13.5 - 6.53i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-2.66 + 11.6i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 0.326iT - 67T^{2} \)
71 \( 1 + (1.48 + 6.49i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (7.02 - 5.60i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 8.81T + 79T^{2} \)
83 \( 1 + (-12.8 + 10.2i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.39 + 6.76i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.52175530273700754857118059081, −10.00775482282648382005267224066, −9.145610175840710538991036351123, −8.028487092302723394905068710654, −7.73703748113061714614280360679, −6.18423316075191208986933119277, −5.34376147647693768180912712037, −4.58101167568380254410154403547, −3.28044568162951975079988979377, −1.62857265079835127876012954055, 1.31874222448963964903981328076, 3.15643946313633220438235077325, 3.82786154544186929419898922887, 4.91560831595415648593464737602, 6.32190886059267496056512234649, 7.11498233618697881301820627956, 7.87932335040442590268203922856, 9.567901647719148359600301727918, 10.06525554671209586637261375135, 10.84863167386687407908357905564

Graph of the $Z$-function along the critical line